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L2 - Response of 1st Order Systems to F(t Governing...

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Response of 1st Order Systems to F(t) Governing equation () dx xF t dt µ += General solution { } . tt xe C eF td t µµ =+ Particular Solutions 1. F ( t ) = K t x KC e t x K

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General case: step function t >= 0, F(t) = 1 0 1 x(t) t t >=0, x(t) = 1 - exp(-t/ µ ) When t µ = , 1 1 1 0.368 0.632 xe =− = Rule o' thumb: Following a stepwise change in forcing, F ( t ), response is 63% complete when t = Corollary: can be measured from step response 2
2. F(t) = A i sin ω t sin( ) x(t) and F(t) are both simple harmonics is constant t o xA t C e µ ωφ ω =− + x t but (some attenuation or gain) and 0 (some phase shift) oi AA φ Solution of pure sine-wave forcing using complex arithmetic Let, () it Ft A e = , then, t xt Ce Be =+ . When F tA e = , then = . Wait a bit!! How did we know that ?? The solution is: iB e B e A e ωω ωµ += Collecting terms and removing the common e i t : ( ) 1 B iA + = So 1 1 B Hi Ai == + This is the transfer function. 3

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How to convert complex transfer function response to amplitude/phase …because, what we observe in the lab is: B H A = H i R Hr Hi φ |H| Clearly, {} 1 22
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L2 - Response of 1st Order Systems to F(t Governing...

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